Facets of congruence distributivity in Goursat categories

Gran, Marino; Rodelo, Diana; Nguefeu, Idriss Tchoffo

doi:10.1016/j.jpaa.2020.106380

Cited by 5 publications

(2 citation statements)

References 18 publications

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“…we may deduce the following ones by applying the quaternary operations p and q, respectively: p(x, x, x, y)T p(x, x, y, y)Rp(x , x , y , y )Sp(x , y , y , y ) We adapt this varietal proof into a categorical one using an appropriate matrix and the corresponding relations which may be deduced from it (see [16] for more details). The kind of matrix we use translates the quaternary identities into the property on relations given in Theorem 4.2(iii):…”

Section: Regular Goursat Categoriesmentioning

confidence: 99%

Geometrical Methods in Goursat Categories

Mbarga

2021

EJMS

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The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, hold in any regular Mal'tsev categories. We prove that Mal'tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, that is classically expressed in terms of four congruences R, S1, S2 and T, and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3. This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2n+1-permutable category $\mathcal{E}$, the category Equiv$(\mathcal{E})$ of equivalence relations in $\mathcal{E}$ is also a 2n+1-permutable category.

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Section: Regular Goursat Categoriesmentioning

confidence: 99%

Geometrical Methods in Goursat Categories

Mbarga

2021

EJMS

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“…In the more general context of regular categories [4] in the place of Barr-exact categories with coequalizers, these become equivalence-distributive Mal'tsev categories in the sense of [19], where the link with strict M -closedness, for the M above, is established. It is easy to prove (and it will be done in Section 5 as an application of our algorithm) that further extension to the left exact context gives us the matrix class of Mal'tsev majority categories, already considered in [22].…”

Section: Matrix Classesmentioning

confidence: 99%

The matrix taxonomy of finitely complete categories

Hoefnagel¹,

Jacqmin²,

Janelidze³

2020

Preprint

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This paper is concerned with the problem of classifying left exact categories according to their 'matrix properties' -a particular category-theoretic property represented by integer matrices. We obtain an algorithm for deciding whether a conjunction of these matrix properties follows from another. Computer implementation of this algorithm allows one to peer into the complex structure of the poset of all 'matrix classes', i.e., the poset of all collections of left exact categories determined by these matrix properties. Among elements of this poset are the collections of Mal'tsev categories, majority categories, (left exact) arithmetical categories, as well as left exact extensions of various classes of varieties of universal algebras, obtained through a process of 'syntactical refinement' of familiar Mal'tsev conditions studied in universal algebra.

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Anticommutativity and the triangular lemma

Hoefnagel

2021

Algebra Univers.

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Facets of congruence distributivity in Goursat categories

Cited by 5 publications

References 18 publications

Geometrical Methods in Goursat Categories

Geometrical Methods in Goursat Categories

The matrix taxonomy of finitely complete categories

Anticommutativity and the triangular lemma

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